On descending cohomology geometrically

Abstract

Abstract. In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur's question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel-Jacobi map admits a distinguished model over the rationals.

Distinguished Models of Intermediate Jacobians

“…Moreover, the arguments via motivated cycles of Section 4 give a second proof of the existence of a phantom abelian variety, although not the descent of the image of the Abel-Jacobi map. In summary, these results strengthen our answer to Mazur's question, given in [ACMV17a].…”

“…Our proof of Theorem A uses a different strategy than we took in [ACMV17a], and as a result improves on the results of that paper in three ways :…”

“…is Aut(C/K)-equivariant, thereby generalizing the well-known cases of the Albanese map A dim X (X C ) → Alb X C and of the Picard map A 1 (X C ) → Pic 0 X C , as well as the case of AJ : A 2 (X C ) → J 3 a (X C ) which was treated in our previous work [ACMV17a]. Theorem A.…”

“…Likewise, one defines regular homomorphisms for smooth projective varieties defined over an arbitrary algebraically closed field. We direct the reader to [ACMV17a, §3] for a review of the material we use here on regular homomorphisms and algebraic representatives, and to [ACMV17a, §4] for the notion of a Galois-equivariant regular homomorphism. In this section we provide some results regarding equivariance of regular homomorphisms ; the main results are Propositions 3.5 and 3.8.…”

A family of cubic fourfolds with finite-dimensional motive

“…(As noted by the referee, one may avoid recourse to [1] here by using the fact that thanks to Collino [10], the Abel-Jacobi map induces an isomorphism from the Albanese of the Fano surface of planes in Z to the intermediate Jacobian of Z. )…”